Algebra Examples

$y = - \frac{1}{2} x^{2} + \frac{14}{5} x + \frac{17}{53}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x^{2}$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2} + \frac{14}{5} x + \frac{17}{53}$

Step 1.1.2

Combine $\frac{14}{5}$ and $x$ .

$y = - \frac{x^{2}}{2} + \frac{14 x}{5} + \frac{17}{53}$

Step 1.2

Complete the square for $- \frac{x^{2}}{2} + \frac{14 x}{5} + \frac{17}{53}$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{2}$

$b = \frac{14}{5}$

$c = \frac{17}{53}$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{14}{5}}{2 (- \frac{1}{2})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{14}{5} \cdot \frac{1}{2 (- \frac{1}{2})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $14$ .

$d = \frac{2 (7)}{5} \cdot \frac{1}{2 (- \frac{1}{2})}$

Step 1.2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 7}{5} \cdot \frac{1}{2 (- \frac{1}{2})}$

Step 1.2.3.2.2.3

Rewrite the expression.

$d = \frac{7}{5} \cdot \frac{1}{- \frac{1}{2}}$

Step 1.2.3.2.3

Multiply $\frac{7}{5}$ by $\frac{1}{- \frac{1}{2}}$ .

$d = \frac{7}{5 (- \frac{1}{2})}$

Step 1.2.3.2.4

Multiply $- 1$ by $5$ .

$d = \frac{7}{- 5 (\frac{1}{2})}$

Step 1.2.3.2.5

Combine $- 5$ and $\frac{1}{2}$ .

$d = \frac{7}{\frac{- 5}{2}}$

Step 1.2.3.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = 7 (\frac{2}{- 5})$

Step 1.2.3.2.7

Move the negative in front of the fraction.

$d = 7 (- \frac{2}{5})$

Step 1.2.3.2.8

Multiply $7 (- \frac{2}{5})$ .

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Step 1.2.3.2.8.1

Multiply $- 1$ by $7$ .

$d = - 7 (\frac{2}{5})$

Step 1.2.3.2.8.2

Combine $- 7$ and $\frac{2}{5}$ .

$d = \frac{- 7 \cdot 2}{5}$

Step 1.2.3.2.8.3

Multiply $- 7$ by $2$ .

$d = \frac{- 14}{5}$

Step 1.2.3.2.9

Move the negative in front of the fraction.

$d = - \frac{14}{5}$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{17}{53} - \frac{{(\frac{14}{5})}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{14}{5}$ .

$e = \frac{17}{53} - \frac{\frac{14^{2}}{5^{2}}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.2

Raise $14$ to the power of $2$ .

$e = \frac{17}{53} - \frac{\frac{196}{5^{2}}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.3

Raise $5$ to the power of $2$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{- 4 (\frac{1}{2})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{\frac{- 4}{2}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $2$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{- 2}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{17}{53} - (\frac{196}{25} \cdot \frac{1}{- 2})$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.5.1

Factor $2$ out of $196$ .

$e = \frac{17}{53} - (\frac{2 (98)}{25} \cdot \frac{1}{- 2})$

Step 1.2.4.2.1.5.2

Factor $2$ out of $- 2$ .

$e = \frac{17}{53} - (\frac{2 \cdot 98}{25} \cdot \frac{1}{2 \cdot - 1})$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = \frac{17}{53} - (\frac{2 \cdot 98}{25} \cdot \frac{1}{2 \cdot - 1})$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = \frac{17}{53} - (\frac{98}{25} \cdot \frac{1}{- 1})$

Step 1.2.4.2.1.6

Multiply $\frac{98}{25}$ by $\frac{1}{- 1}$ .

$e = \frac{17}{53} - \frac{98}{25 \cdot - 1}$

Step 1.2.4.2.1.7

Multiply $25$ by $- 1$ .

$e = \frac{17}{53} - \frac{98}{- 25}$

Step 1.2.4.2.1.8

Move the negative in front of the fraction.

$e = \frac{17}{53} - - \frac{98}{25}$

Step 1.2.4.2.1.9

Multiply $- - \frac{98}{25}$ .

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Step 1.2.4.2.1.9.1

Multiply $- 1$ by $- 1$ .

$e = \frac{17}{53} + 1 (\frac{98}{25})$

Step 1.2.4.2.1.9.2

Multiply $\frac{98}{25}$ by $1$ .

$e = \frac{17}{53} + \frac{98}{25}$

Step 1.2.4.2.2

To write $\frac{17}{53}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$e = \frac{17}{53} \cdot \frac{25}{25} + \frac{98}{25}$

Step 1.2.4.2.3

To write $\frac{98}{25}$ as a fraction with a common denominator, multiply by $\frac{53}{53}$ .

$e = \frac{17}{53} \cdot \frac{25}{25} + \frac{98}{25} \cdot \frac{53}{53}$

Step 1.2.4.2.4

Write each expression with a common denominator of $1325$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.2.4.1

Multiply $\frac{17}{53}$ by $\frac{25}{25}$ .

$e = \frac{17 \cdot 25}{53 \cdot 25} + \frac{98}{25} \cdot \frac{53}{53}$

Step 1.2.4.2.4.2

Multiply $53$ by $25$ .

$e = \frac{17 \cdot 25}{1325} + \frac{98}{25} \cdot \frac{53}{53}$

Step 1.2.4.2.4.3

Multiply $\frac{98}{25}$ by $\frac{53}{53}$ .

$e = \frac{17 \cdot 25}{1325} + \frac{98 \cdot 53}{25 \cdot 53}$

Step 1.2.4.2.4.4

Multiply $25$ by $53$ .

$e = \frac{17 \cdot 25}{1325} + \frac{98 \cdot 53}{1325}$

Step 1.2.4.2.5

Combine the numerators over the common denominator.

$e = \frac{17 \cdot 25 + 98 \cdot 53}{1325}$

Step 1.2.4.2.6

Simplify the numerator.

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Step 1.2.4.2.6.1

Multiply $17$ by $25$ .

$e = \frac{425 + 98 \cdot 53}{1325}$

Step 1.2.4.2.6.2

Multiply $98$ by $53$ .

$e = \frac{425 + 5194}{1325}$

Step 1.2.4.2.6.3

Add $425$ and $5194$ .

$e = \frac{5619}{1325}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - \frac{14}{5})}^{2} + \frac{5619}{1325}$ .

$- \frac{1}{2} {(x - \frac{14}{5})}^{2} + \frac{5619}{1325}$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x - \frac{14}{5})}^{2} + \frac{5619}{1325}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{2}$

$h = \frac{14}{5}$

$k = \frac{5619}{1325}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{14}{5}, \frac{5619}{1325})$

Step 4

Find $p$ , the distance from the vertex to the focus.

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Step 4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Step 4.3.2

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Step 4.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

Step 5

Find the directrix.

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Step 5.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 5.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{12563}{2650}$

Step 6